We may, however, extend the construction just given, even to those arches that are formed of portions of cir cles differing in curvature. For the equilibrating ex trados being first constructed for that portion of the arch in which the crown is, as far as the vertical line passing through the contact of the neighbouring curves, the thickness of the crown must be supposed to be enlarged, in proportion to the diminution of the radius of curva ture, or the contrary, and, with this, proceed as before along the succeeding branch of the curve. This will, indeed, cause an unsightly break in the extrados, for which we shall not at present pretend to find any other remedy, than using materials of a different specific gravity.
Those who wish to examine this subject farther, may consult Emerson's Fluxions, or lIutton's Principles of Bridges. We shall only observe here, that the ex trados of the ellipse, and of the cycloid, resemble that of the circle, having an infinite arc on each side at the springing ; and indeed this, as has already been observ ed, is a general rule for all those curves which spring at right angles to the horizon. In the parabola, the extrados is another parabola exactly the same, only removed a little above the other. In the hyperbola, the is another curve, which approaches the interior arch towards the springing. None of these curves, therefore, can, with propriety, be employed for the arches of a bridge, though there may be cases where a single arch might with propriety be formed into a conic section.
The catenaria, which has been much spoken of as the best form for an arch, has an extrados, the depression of which, below its crown, at any point, is to the depres sion of the curve in the same vertical line, in a constant ratio. This ratio is that of the constant tension at the vertex, to the same tension diminished by the thickness or vertical pressure in the crown. If the vertical pres sure be less than the tension, the extrados falls below the horizontal line ; if greater, it will rise above it.
Mathematicians finding the circle and other common curves so little adapted to the arch of a bridge, which has a horizontal roadway, have, in the next place, endea voured to solve the converse of the problem, and give a rule for finding the intrados or figure of the arch, which have the exterior curve a horizontal line.
This problem can only be resolved by calling the fluxionary calculus to our aid. It is a case of the more general one to find the intrados, when the extrados is given ; and being the most useful case of that problem, fortunately admits of a solution comparatively easy.
We subjoin a Table, calculated by Dr Mutton from this formula, for an arch of 100 feet span and 40 feet rise, the thickness of the crown being taken at 6 feet. It is nearly of the same dimensions as the middle arch of Blackfriar's Bridge, and which may answer for any arch where these dimensions are similarly related to each other.
The curve of Fig. 8. Plate LXXX. is accurately drawn to these dimensions, and may give an idea of the form of an equilibrated arch. It is not destitute of grace, and is abundantly roomy for craft.
Such, then, is the analytical theory of equilibration ; for a practical subject it does, we confess, appear abstruse.
Those who have already studied the theory, will ob serve, that we have greatly simplified the investigation. The construction we have given for circular arches we shall probably find useful hereafter. We could with pleasure have prosecuted the subject farther, not only as it affords some good general views of the equilibration of arches, but exhibits also several beautiful examples of the application of the higher calculus. Yet we must repeat, with all due respect to the learned and eminent men who have turned their attention to it, that we fear their speculations have been of little value. In saying this, we do not mean to surmise, that their deductions are any way erroneous ; they are legitimate consequences from the principles assumed. But it appears to us, that the writers on equilibration, like many others who have hastily applied analysis to physics, have taken too nar row a view of their subject to comprehend all the variety of practice. Setting out with one leading prin ciple, best adapted, perhaps, to the application of cal culus, they neglect the numerous circumstances by which it may be modified, and which are too important to be overlooked in drawing practical inferences from such an investigation.
Their principal care respects the figure of the soffit, a thing which the practical engineer knows may admit of the greatest variety. As to the thickness of arch stones, side wall, and piers, the horizontal section or ground plan of the bridge, the manner of filling up its haunches, of forming the joints, of connecting it with the abutments, wing walls, &c. the are still left in the dark.